The local smoothing conjecture implies several open problems in harmonic analysis, like the Bochner–Riesz conjecture and the restriction conjecture. A breakthrough result was the proof of the sharp \(\ell ^2\) -decoupling inequality for the cone, due to Bourgain–Demeter [1]}. More precisely, set
\(\overline{s}(p):={\left\lbrace \begin{array}{ll}0,& 2 \le p \le \frac{2(d+1)}{d-1},\\s(p) - \frac{1}{p},&\frac{2(d+1)}{d-1} \le p < \infty .\end{array}\right.}\) 
